A new energy drink is being tested for its effect on alertness. In a study, $160$ participants consumed the drink, and $28$ reported increased alertness. Someone claims that more than $12\%$ of users experience increased alertness. A hypothesis test is run at a $0.05$ significance level. If the true proportion of users experiencing increased alertness is $0.20$, and the power of the test is $0.90$, how should this power be interpreted?

A researcher analyzes the last digits of lottery ticket numbers in a city and finds that among $1500$ randomly selected digits, $175$ are sevens. If the digits are randomly distributed, the proportion of sevens should be $0.1$. Use the critical value method with a $0.05$ significance level to test the claim that the proportion of sevens equals $0.1$. What is your conclusion?

A car manufacturer claims the average fuel efficiency of their new model is $40$ miles per gallon. A sample test yields a $t$-statistic of $1.60$. At the $0.05$ significance level, should the null hypothesis be rejected in a two-tailed test?

A scientist wants to determine whether a new fertilizer increases the growth rate of tomato plants compared to the standard fertilizer. After the experiment, the $p$-value is calculated to be $0.22$. At a significance level of $0.05$, what should the scientist conclude about the null hypothesis and the effectiveness of the new fertilizer?

A pharmaceutical company wants to determine if their new drug reduces cholesterol more effectively than a placebo. The reduction in cholesterol in milligrams per deciliter was measured after three months. The data are as follows: 

Assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. Use degrees of freedom equal to the smaller of $n_1-1$ and $n_2-1$. At the $0.05$ significance level, test the claim that the new drug leads to a greater mean reduction in cholesterol than the placebo. 

In a statistical test, the calculated test statistic is $t=1.421$. Does this value indicate that you should reject the null hypothesis?

Find the critical value and rejection region for a left-tailed $z$-test where $\alpha =0.025$.